{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw3 - Modern Analysis Homework 3 1 Recall that a function f...

This preview shows pages 1–2. Sign up to view the full content.

Modern Analysis, Homework 3 1. Recall that a function f on an interval [ a, b ] is of bounded variation if sup n N x 0 = a<x 1 < ··· <x n = b n X i =0 | f ( x i ) - f ( x i +1 ) | < + . In such a case, we write f BV[ a, b ]. Given f BV[ a, b ], for x [ a, b ], define T f ( x ) = sup n N x 0 = a<x 1 < ··· <x n = x n X i =0 | f ( x i ) - f ( x i +1 ) | (the total variation of f on [ a, x ]). (a) Prove that T f ( x ) + f ( x ) and T f ( x ) - f ( x ) are increasing functions on [ a, b ]. (b) Prove that f BV[ a, b ] if and only if f can be written as a difference of increasing functions. 2. Let { f n } n N be a sequence of continuous functions defined on a compact metric space ( K, d ). Suppose that for every convergent sequence of points { x n } n N such that x n x , we have lim n →∞ f n ( x n ) = f ( x ) where f is some continuous function on K . Prove that f n f uniformly on K . 3. Let g be a continuous function on [0 , 1] with g (1) = 0. Prove that the sequence { g ( x ) x n } converges uniformly on [0 , 1]. 4. (a) Suppose that f is a 2 π -periodic function on R , Riemann-integrable on [ - π, π ]. Let ˆ f ( n ) denote the n -th Fourier coefficient of f (ie ˆ f ( n ) = 1 2 π R π - π f ( x ) e - inx dx ), and suppose that ˆ f ( n ) = 0 for all n Z . Prove that f ( x 0 ) = 0 whenever f

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

hw3 - Modern Analysis Homework 3 1 Recall that a function f...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online